Differential Operators on Supercircle: Conformally Equivariant Quantization and Symbol Calculus

Gargoubi, Hichem; Mellouli, Najla; Ovsienko, Valentin

doi:10.1007/s11005-006-0129-8

Cited by 58 publications

(76 citation statements)

References 21 publications

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“…Then the Killing form of sl(p + 1|q) is given by K(A, B) = str(ad(A)ad(B)) = 2(p + 1 − q) str(AB) (13) for all A and B in sl(p + 1|q). Under our assumption q = p + 1, it is a non-degenerate even supersymmetric bilinear form on this algebra.…”

Section: Casimir Operatorsmentioning

confidence: 99%

“…Recently, several papers dealt with the problem of equivariant quantizations in the context of supergeometry: first, in [13,30] the problem of equivariant quantizations over the supercircles S 1|1 and S 1|2 was considered (with respect to orthosymplectic superalgebras). Second, the thesis [31] dealt with conformally equivariant quantizations over supercotangent bundles.…”

Section: Introductionmentioning

confidence: 99%

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Projectively Equivariant Quantizations over the Superspace $${\mathbb{R}^{p|q}}$$

Mathonet

Radoux

2011

Lett Math Phys

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Abstract. We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of pgl(n|n) ∼ = sl(n|n)), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields. MSC(2010): 17B66, 58A50

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Section: Casimir Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Projectively Equivariant Quantizations over the Superspace $${\mathbb{R}^{p|q}}$$

Mathonet

Radoux

2011

Lett Math Phys

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“…One checks that g(x)D acts on F (λ) by the differential operator gD + 2λξg ′ . In this context, the fine filtration was introduced in [GMO07]. It is the Con R 1|1 -invariant N/2-filtration The explicit formula for CQ λ,p was deduced in [CMZ97] for (λ, p) = (0, 0), and in [GMO07] in general.…”

Section: |1mentioning

confidence: 99%

“…In this context, the fine filtration was introduced in [GMO07]. It is the Con R 1|1 -invariant N/2-filtration The explicit formula for CQ λ,p was deduced in [CMZ97] for (λ, p) = (0, 0), and in [GMO07] in general. Problems 1 and 5 are largely solved in [Co08], and Problem 4 was reduced to computation.…”

Section: |1mentioning

confidence: 99%

Symmetry in Mathematics and Physics

Conley¹

2009

Contemporary Mathematics

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“…Let us now recall the following classic result [17,13]: if X ∈ k(1), there exists a unique superfunction f (x, ξ) = a(x) − 2ξb(x), called the contact Hamiltonian, such that X = X f , where…”

Section: The Super Lie Algebra K(1) Of Contact Vector Fieldsmentioning

confidence: 99%

On the Projective Geometry of the Supercircle: A Unified Construction of the Super Cross-Ratio and Schwarzian Derivative

Michel

Duval

2008

International Mathematics Research Notices

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On the projective geometry of the supercircle: a unified construction of the super cross-ratio and Schwarzian derivative J.-P. Michel ‡ C. Duval § Centre de Physique Théorique, CNRS, Luminy, Case 907 F-13288 Marseille Cedex 9 (France) ¶ Abstract We consider the standard contact structure on the supercircle, S 1|1 , and the supergroups E(1|1), Aff(1|1) and SpO(2|1) of contactomorphisms, defining the Euclidean, affine and projective geometry respectively. Using the new notion of p|q-transitivity, we construct in synthetic fashion even and odd invariants characterizing each geometry, and obtain an even and an odd super cross-ratios.Starting from the even invariants, we derive, using a superized Cartan formula, one-cocycles of the group of contactomorphisms, K(1), with values in tensor densities F λ (S 1|1 ). The even cross-ratio yields a K(1) one-cocycle with values in quadratic differentials, Q(S 1|1 ), whose projection on F 3 2 (S 1|1 ) corresponds to the super Schwarzian derivative arising in superconformal field theory. This leads to the classification of the cohomology spacesThe construction is extended to the case of S 1|N . All previous invariants admit a prolongation for N > 1, as well as the associated Euclidean and affine cocycles. The super Schwarzian derivative is obtained from the even cross-ratio, for N = 2, as a projection to F 1 (S 1|2 ) of a K(2) one-cocycle with values in Q(S 1|2 ). The obstruction to obtain, for N ≥ 3, a projective cocycle is pointed out. Keywords IntroductionThe cross-ratio is the fundamental object of projective geometry; it is a projective invariant of the circle S 1 (or, rather, of RP 1 ). The main objective of this article is to propose and justify from a group theoretical analysis a super-analogue of the cross-ratio in the case of the supercircle S 1|N , and to deduce then, from the Cartan formula (1.2), the associated Schwarzian derivative for N = 1, 2.It is well-known that the circle, S 1 , admits three different geometries, namely the Euclidean, affine and projective geometries, as highlighted by Ghys [14]. They are defined by the groups (R, +), Aff(1, R) and PGL(2, R), or equivalently by their characteristic invariants, the distance, the distance ratio, and the cross-ratio. From these invariants we can obtain, using Cartan-like formulae, three 1-cocycles of Diff + (S 1 ) with coefficients in some tensorial density modules F λ (S 1 ) with λ ∈ R; see [9]. They are the generators of the three nontrivial cohomology spaces H 1 (Diff + (S 1 ), F λ ), with λ = 0, 1, 2, as proved in [12].The purpose of this article is to extend these results to the supercircle, S 1|N , en- Quite independently, and from a more mathematical point of view, Manin [23] introduced the even and odd cross-ratios, for N = 1, 2, by resorting to linear super-

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Differential Operators on Supercircle: Conformally Equivariant Quantization and Symbol Calculus

Cited by 58 publications

References 21 publications

Projectively Equivariant Quantizations over the Superspace $${\mathbb{R}^{p|q}}$$

Projectively Equivariant Quantizations over the Superspace $${\mathbb{R}^{p|q}}$$

Symmetry in Mathematics and Physics

On the Projective Geometry of the Supercircle: A Unified Construction of the Super Cross-Ratio and Schwarzian Derivative

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